Lecture Notes for Analog Electronics - The Electronic Universe ...

of interest are actually in a narrow band centered about the carrier frequency. Without the
diode, the system is linear, and no signal will be present at the output.
The diode is non-linear; recall its V -I curve. In order to illustrate how this works, we
assume a specific form for the response of a forward-biased diode as I = bV 2 ,wherebis a
constant. A resistor r is inserted between point a and ground (Fig. 44) in order to convert
this diode current to a voltage to be presented to the low-pass filter. Now let V be the linear
combination of two signals: V = V1 cos ω1t + V2 cos ω2t. This then gives rise to an output
current
I = bV 2
1 cos 2 ω1t + bV 2
2 cos 2 ω2t +2bV1V2 cos ω1t cos ω2t
Again using trigonometric identities to form the poor man’s Fourier transform, this becomes
2I/b = V 2
1
+ V 2
2
+ V 2
1 cos 2ω1t + V 2
2 cos 2ω2t +2V1V2[cos((ω1 + ω2)t)+cos((ω1−ω2)t)]
Therefore, from the original two frequencies, the diode has produced harmonics (twice the
original), as well as the sum and difference.
In the case of our simplified AM broadcast signal of Eqn. 46, where three frequencies
are originally present (ωc and ωc ± ωm), the effect of the diode is easily generalized from the
steps above using the substitutions ω1 = ωc and ω2 = ωc + ωm or ω2 = ωc − ωm. We find
that the output of the diode will include DC, the first harmonics of all three frequecies, as
well as the six possible sum and difference frequencies. Of particular interest for our receiver
is the difference frequency between the carrier and the modulated carrier. For example,
ωc − (ωc − ωm) =ωm
Therefore, we do in fact recover a Fourier component corresponding to our original modulating
signal. This can then be separated from the higher frequencies using the low-pass filter
and amplifier. This represents a simple example of so-called heterodyne detection, inwhich
different frequencies are combined in order to extract a difference frequency.
As an aside, we note that with our example I = bV 2 , we have squared the input. When
we examine this in frequency domain (Fourier transform) and low-pass filter the result (averaging),
we have effectively formed the so-called power spectrum of the input, also called
the spectral power density.
8.3.2 Harmonic Distortion
Note that we intentionally introduced a non-linear element (the diode) to our system. An unintentional
non-linearity in a circuit, for example in an audio amplifier circuit, can introduce
additional frequencies as demonstrated above. In particular, our diode with the I = bV 2 behavior
introduced first harmonics of the original frequencies at twice the original. In general,
a non-linearity may include any number of higher-order terms: I = b1V + b2V 2 + b3V 3 + ···,
where each additional power can generate the next higher harmonic. For example, a nonzero
b3 will generate a 2nd harmonic of the original ω at 3ω. The introduction of harmonics
of the input signal is called harmonic distortion. Since the pattern of harmonics is what
distinguishes musical instrument types to the ear, the introduction of non-linearities should
be avoided in high-fidelity audio amplifiers.
8.3.3 Homodyne Detection
An example of this technique is given in the text, pages 653 and 889. It uses a phase-locked
loop (PLL) circuit at the input of the receiver. Recall that the PLL circuit is designed
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